critical thinking and problem solving in mathematics

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Free ready-to-use math resources

Hundreds of free math resources created by experienced math teachers to save time, build engagement and accelerate growth

20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE :

• 8 Common Core math examples
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies for problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

Related articles

Why Student Centered Learning Is Important: A Guide For Educators

13 Effective Learning Strategies: A Guide to Using them in your Math Classroom

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

5 Math Mastery Strategies To Incorporate Into Your 4th and 5th Grade Classrooms

Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

Privacy Overview

How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

• Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
• Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
• Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
• Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

• Identification: Recognize and define the problem.
• Analysis: Understand the problem’s intricacies and nuances.
• Generation of Alternatives: Think of different ways to approach the challenge.
• Decision Making: Choose the most suitable method to address the problem.
• Implementation: Put the chosen solution into action.
• Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

• Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
• Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
• Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

More than Numbers 

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

Related posts

Summer Math Programs: How They Can Prevent Learning Loss in Young Students

As summer approaches, parents and educators alike turn their attention to how they can support young learners during the break. Summer is a time for relaxation, fun, and travel, yet it’s also a critical period when learning loss can occur. This phenomenon, often referred to as the “summer slide,” impacts students’ progress, especially in foundational subjects like mathematics. It’s reported…

Math Programs 101: What Every Parent Should Know When Looking For A Math Program

  As a parent, you know that a solid foundation in mathematics is crucial for your child’s success, both in school and in life. But with so many math programs and math help services out there, how do you choose the right one? Whether you’re considering Outschool classes, searching for “math tutoring near me,” or exploring tutoring services online, understanding…

• Math Resources Links
• Math in the Real World
• Differentiated Math Unlocked
• Math in the Real World Workshop

20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

The level of apathy towards math is only increasing as each year passes and it’s up to us as teachers to make math class more meaningful . This list of math critical thinking questions will give you a quick starting point for getting your students to think deeper about any concept or problem. 

Since artificial intelligence has basically changed schooling as we once knew it, I’ve seen a lot of districts and teachers looking for ways to lean into AI rather than run from it.

The idea of memorizing formulas and regurgitating information for a test is becoming more obsolete. We can now teach our students how to use their resources to make educated decisions and solve more complex problems.

With that in mind, teachers have more opportunities to get their students thinking about the why rather than the how.

Table of Contents

Looking for more about critical thinking skills? Check out these blog posts:

• Why You Need to Be Teaching Writing in Math Class Today
• How to Teach Problem Solving for Mathematics
• Turn the Bloom’s Taxonomy Verbs into Engaging Math Activities

What skills do we actually want to teach our students?

As professionals, we talk a lot about transferable skills that can be valuable in multiple jobs, such as leadership, event planning, or effective communication. The same can be said for high school students. 

It’s important to think about the skills that we want them to have before they are catapulted into the adult world. 

Do you want them to be able to collaborate and communicate effectively with their peers? Maybe you would prefer that they can articulate their thoughts in a way that makes sense to someone who knows nothing about the topic.

Whatever you decide are the most essential skills your students should learn, make sure to add them into your lesson objectives.

When should I ask these math critical thinking questions?

Critical thinking doesn’t have to be complex or fill an entire lesson. There are simple ways that you can start adding these types of questions into your lessons daily!

Start small

Add specific math critical thinking questions to your warm up or exit ticket routine. This is a great way to start or end your class because your students will be able to quickly show you what they understand. 

Asking deeper questions at the beginning of your class can end up leading to really great discussions and get your students talking about math.

Add critical thinking questions to word problems

Word problems and real-life applications are the perfect place to add in critical thinking questions. Real-world applications offer a more choose-your-own-adventure style assignment where your students can expand on their thought processes. 

They also allow your students to get creative and think outside of the box. These problem-solving skills play a critical role in helping your students develop critical thinking abilities.

Keep reading for math critical thinking questions that can be applied to any subject or topic!

When you want your students to defend their answers.

• Explain the steps you took to solve this problem
• How do you know that your answer is correct?
• Draw a diagram to prove your solution.
• Is there a different way to solve this problem besides the one you used?
• How would you explain _______________ to a student in the grade below you?
• Why does this strategy work?
• Use evidence from the problem/data to defend your answer in complete sentences.

When you want your students to justify their opinions

• What do you think will happen when ______?
• Do you agree/disagree with _______?
• What are the similarities and differences between ________ and __________?
• What suggestions would you give to this student?
• What is the most efficient way to solve this problem?
• How did you decide on your first step for solving this problem?

When you want your students to think outside of the box

• How can ______________ be used in the real world?
• What might be a common error that a student could make when solving this problem?
• How is _____________ topic similar to _______________ (previous topic)?
• What examples can you think of that would not work with this problem solving method?
• What would happen if __________ changed?
• Create your own problem that would give a solution of ______________.
• What other math skills did you need to use to solve this problem?

Let’s Recap:

• Rather than running from AI, help your students use it as a tool to expand their thinking.
• Identify a few transferable skills that you want your students to learn and make a goal for how you can help them develop these skills.
• Add critical thinking questions to your daily warm ups or exit tickets.
• Ask your students to explain their thinking when solving a word problem.
• Get a free sample of my Algebra 1 critical thinking questions ↓

8 thoughts on “20 Math Critical Thinking Questions to Ask in Class Tomorrow”

I would love to see your free math writing prompts, but there is no place for me to sign up. thank you

Ahh sorry about that! I just updated the button link!

Pingback:  How to Teach Problem Solving for Mathematics -

Pingback:  5 Ways Teaching Collaboration Can Transform Your Math Classroom

Pingback:  3 Ways Math Rubrics Will Revitalize Your Summative Assessments

Pingback:  How to Use Math Stations to Teach Problem Solving Skills

Pingback:  How to Seamlessly Add Critical Thinking Questions to Any Math Assessment

Pingback:  13 Math Posters and Math Classroom Ideas for High School

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Check Out the New Website Shop!

Novels & Picture Books

Anchor Charts

6 Strategies for Increasing Critical Thinking with Problem Solving

By Mary Montero

Share This Post:

• Facebook Share
• Twitter Share
• Pinterest Share
• Email Share

For many teachers, problem-solving feels synonymous with word problems, but it is so much more. That’s why I’m sharing my absolute favorite lessons and strategies for increasing critical thinking through problem solving below. You’ll learn six strategies for increasing critical thinking through mathematical word problems, the importance of incorporating error analysis into your weekly routines,  and several resources I use for improving critical thinking – almost all of which are free! I’ll also briefly touch on teaching students to dissect word problems in a way that enables them to truly understand what steps to take to solve the problem.

This post is based on my short and sweet (and FREE!) Increasing Critical Thinking with problem Solving math mini-course . When you enroll in the free course you’ll get access to everything you need to get started:

• Problem Solving Essentials
• Six lessons to implement into your classroom
• How to Implement Error Analysis
• FREE Error Analysis Starter Kit
• FREE Mathematician Posters
• FREE Multi-Step Problem Solving Starter Kit
• FREE Task Card Starter Kit

Introduction to Critical Thinking and Problem Solving

According to the National Council of Teachers of Mathematics, “The term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development .)”

That’s a lot of words, but I’d like to focus in on the word POTENTIAL. I’m going to share with you strategies that move these tasks from having the potential to provide a challenge to actually providing that challenge that will enrich their mathematical understanding and development. 

If you’re looking for an introduction to multi-step problem solving, I have a free multi-step problem solving starter kit for that! 

I also highly encourage you to download and use my free Mathematician posters that help students see what their “jobs” are as mathematicians. Giving students this title of mathematician not only holds them accountable, but it gives them greater confidence and gives me very specific verbiage to use when discussing math with my students. 

The impacts of Incorporating Problem Solving

When I made the shift to incorporate problem solving into my everyday instruction intentionally, I saw a distinct increase in student understanding and application of mathematical concepts, more authentic connections to real-world mathematics scenarios, greater student achievement, and notably increased engagement. There are also ripple effects observed in other areas, as students learn grit and a growth mindset after tackling some more challenging problem-solving situations. I hope that by implementing some of these ideas, you see the very same shift.

Here’s an overview of some problem solving essentials I use to teach students to solve problems.

Routine vs. Non-Routine Problem Solving

Routine problems comprise the vast majority of the word problems we pose to students. They require using an algorithm through one or more of the four major operations, have relevance to real-world situations, and often have a distinct answer. They are solvable, and students can use several concrete strategies for solving, like “make a table” or “draw a picture” to solve.

Conversely, non-routine problem-solving focuses on mathematical reasoning. These are often more open-ended and allow students to make generalizations about math and numbers. There isn’t usually a straight path leading to the answer, there isn’t an algorithm readily available for finding the solution (or students are going to have to come up with the algorithm), and it IS going to require some level of experimentation and manipulation of numbers in order to solve it. In non-routine problems, students learn to look for patterns, work backwards, build models, etc. 

Incorporating both routine and non-routine problems into your instruction for EVERY student is critical. When solving non-routine problems, students can use some of the strategies they’ve learned for solving routine problems, and when solving routine problems, students benefit from a deeper understanding of the complexity of numbers that they gained from non-routine problems. For this training, we will focus heavily on routine problems, though the impacts of these practices will transition into non-routine problem solving.

Increasing Critical Thinking in Problem Solving

When tackling a problem, students need to be able to determine WHAT to do and HOW to do it.  Knowing the HOW is what you likely teach every day – your students know how to add, subtract, multiply, and divide. But knowing WHAT to do is arguably the most essential part of solving problems – once students know what needs to be done, then they can apply the conceptual skills – the algorithms and strategies – they’ve learned and will know how to solve. While dissecting word problems is an excellent starting point, exposing students to various ways to examine problems can help them figure out the WHAT. 

Being faced with a lengthy, complex word problem can be intimidating to even your most adept students. Having a toolbox of strategies to use when you tackle problems and seeing problems in various ways can enable students to get to the point where they feel comfortable knowing where to begin.

Shifting away from keywords

While it isn’t best practice to rely solely on operation “keywords” to determine what operation needs to occur when solving a problem, I’m not ready to fully ditch keyword-based instruction in math. I think there’s a huge difference between teaching students to blindly rely on keywords to determine which operation to use for a solution and using words found in the text to guide students in figuring out what to do. For that reason, I place heavy emphasis on using precise mathematical vocabulary , including specific operation keywords, and when students become accustomed to using that precise mathematical vocabulary every day, it really helps them to identify that language in word problems as well.

I also allow my students to dissect math word problems using strategies like CUBES , but in a way that is more aligned with best practice. 

Six Lessons for Easy Implementation

Here are six super quick “outside the box” word problem, problem solving lessons to begin implementing into your classroom. These lessons shouldn’t replace your everyday problem solving, but are instead extensions that will help students tackle those tricky problems they encounter everyday. As a reminder, we look at all of these lessons in the FREE Increasing Critical Thinking with problem Solving math mini-course .

Lesson #1: What’s the Question?

In this lesson, we’ll encourage students to see. just how many different questions can be asked about the same statements or information. We start with a typical, one-step, one-operation problem. Then we cross out or cover up the answer and ask students to generate possible questions.

After students have come up with a variety of questions, ask them to determine HOW they would solve for each one.

Reveal the question and ask students how they would solve this one and see if any of the questions they came up with match.

This activity is important because it demonstrates to students just how many different questions can be asked about the same statement or information. It’s perfect for your students who automatically pick out numbers and start “operating” on them blindly. I’ve had students come up with 5-8 questions with a single statement!

I like to do this throughout the year using different word problems based on the skill we’re focused on at the time AND skills we’ve previously mastered, but be careful not to only use examples based on the skill you’re teaching right then so their brains don’t automatically go to the same place.

These 32 What’s My Operation? task cards will help your student learn and review which operations to use for different types of word problems! They’re perfect to use as a quick assessment, game of SCOOT, math center activity, or homework.

Lesson 2: Similar Scenarios

In this lesson, students will evaluate similar scenarios to determine the appropriate operations. Start with three similar scenarios requiring different operations and identify what situation is happening in each scenario (finding total, determining an amount, splitting or combining, etc.).

Read all three-word problems on a similar topic. Determine the similarity of all of them and determine which operation would be used to solve them. How does the situation/action of the problem help you determine what step to take?

I also created these differentiated word problem task cards after noticing my students struggling with which operation to choose, especially when given multiple problems from a similar scenario. They encourage students to select the appropriate operation for each word problem.

Lesson 3: Opposing Operations

In this lesson, students will determine relevant information from a set of facts, which requires a great deal of critical thinking to determine which operation to use. Give students a scenario and a variety of facts/information relating to the scenario as well as several questions to answer based on the facts . Students will focus on determining HOW they will solve each question using only the relevant information. 

These Operation Fascination task c ards engage students in critical thinking about operations. Each card has a scenario, multiple clues and facts to support the scenario, and four questions to accompany each scenario. The questions are a variety of operations so that students can see how using the same information can solve multiple problems.

Lesson 4: Next Level Numberless

In this lesson, we’ll take numberless word problems to the next level by developing a strong conceptual understanding of word problems. Give students scenarios without numbers and have them write a question and/or insert numbers using a specific operation and purpose . This requires a great deal of thinking to not only determine the situation, but to also figure out numbers that fit into the situation in a way that makes sense.

By integrating these types of math problems into your daily lessons, you can significantly enhance your students’ comprehension of word problems and problem-solving. These numberless word problem task cards are the ideal to improve your students’ critical thinking and problem-solving skills. They offer a variety of numbered and numberless word problems.

Lesson 5: Story Situations

In this lesson, we’ll discuss the importance of students generating their own word problems with a given set of information. This requires a great deal of quantitative reasoning as students determine how they would use a given set of numbers to create a realistic situation. Present students with two predetermined numbers and a theme. Then have students write a word problem, including a question, using the given information. 

Engage your students in additional practice with these differentiated division task cards that require your students to write their OWN word problems (and create real-world relevance in their learning!). Each task card has numbers and a theme that students use to guide their thinking and creation of a word problem.

Lesson 6: No Scenario Solving

In this lesson, we’ll decontextualize problem solving and require students to create the situation, represent it numerically, and solve. It’s a cognitively demanding task! Give students an operation and a purpose (joining, separating, comparing, etc.) with no other context, numbers, numbers, or theme. Then have students generate a word problem.

For additional practice, have students swap problems to identify the operation, purpose, and solution.

Implementing Error Analysis

Error analysis is an exceptional way to promote thinking and learning, but how do we teach students to figure out which type of math error they’ve made? This error analysis starter kit can help!

First, it is very rare that I will tell my students what error they have made in their work. I want to challenge them to figure it out on their own. So, when I see that they have a wrong answer, I ask them to go back and figure out where something went wrong. Because I resist the urge to tell them right away where their error is, my students tend to get a lot more practice identifying them!

Second, when I introduce a concept, I always, always, always create anchor charts with students and complete interactive notebook activities with them so that they have step-by-step procedures for completing tasks right at their fingertips. I have them go back and reference their notebooks while they are looking at their errors.  Usually, they can follow the anchor chart step-by-step to make sure they haven’t made a conceptual error, and if they have, they can identify it.

Third, I let them use a calculator. When worst comes to worst, and they are fairly certain they haven’t made a conceptual mistake to identify, I let them get out a calculator and start computing, step-by-step to see where they’ve made a mistake.

IF, after taking these steps, a student can’t figure out their mistake (especially if I find that it’s a conceptual mistake), I know I need to go back and do some individual reteaching with them because they don’t have a solid understanding of the concept.

This FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

Digging Deeper into Error Analysis

Once students show proficiency in the standard algorithm (or strategies), I take it a step further and have them dive into error analysis where they can show a “reverse” understanding as they evaluate mistakes made and fix them. Being able to identify an error in someone else’s work requires higher order thinking not found in most other projects or activities and certainly not found in basic math fact completion.

First, teach students the difference between a computational error and a conceptual error. 

• Computational is when they make a mistake in basic math facts. This might look as simple as  64/8 does not equal 7. Oops!
• A Conceptual or Procedural Error is when they make a mistake in the procedure or concept. 
• I can’t tell you how many times students show as not proficient on a topic when the mistakes they are making are COMPUTATIONAL and not conceptual or procedural. They don’t need more review in how to use a strategy… they need to slow down and pay closer attention to their math facts!

Once we’ve introduced the types of errors they should be looking out for, we move on to actually analyzing these errors in someone else’s work and fixing the mistake.

I have created error analysis tasks for you to use with you students so they can identify the errors, types of errors, rework the problem, and create their own version of the problem and solve it. I have seen great success with incorporating these tasks into ALL of my math units. I even have kids beg to take their error analysis tasks out to recess to finish! These are great resources to start:

• Error Analysis Bundle
• 3rd Grade Word Problem of the Day
• 4th Grade Word Problem of the Day
• 5th Grade Word Problem of the Day

The final step in using error analysis is actually having students correct their OWN mistakes. Once I have instructed on types of errors, I will start by simply telling them, Oops! You’ve made a computational error here! That way they aren’t furiously looking through the procedure for a mistake, instead they are looking to see where they computed wrong. Conversely, I’ll tell them if they’ve made a procedural mistake, and that can guide them in figuring out what they need to look for.

Looking at the different types of errors students are making is essential to guiding my instruction as well, so even though it takes a bit longer to grade things like this, it is immensely helpful to me as I make adjustments to my instruction.

Resources and Ideas for Critical Thinking

I’ve compiled a collection of websites for complex tasks with multiple, open-ended answers and scenarios. The majority of these tasks are non-routine and so easy to implement. I often post these tasks and allow students short bursts of time to strategize and plan for a solution. Consider using the tasks and problems from these sites as warm-ups, extensions of your morning meeting, during enrichment groups, or on a Problem of the Week board. I also highly encourage you to incorporate these non-routine problems into your core instruction time for all students at least once or twice a month.

• NRICH provides thousands of FREE online mathematics resources for ages 3 to 18. The tasks focus on developing problem-solving skills, perseverance, mathematical reasoning, the ability to apply knowledge creatively in unfamiliar contexts, and confidence in tackling new challenges..
• Open Middle offers challenging math word problems that require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking. All problems have a “closed beginning,” meaning that they all start with the same initial problem, a “closed-end” meaning that they all end with the same answer, and an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
• Mathcurious offers interactive digital puzzles. Each adventure is dedicated to exploring the world of math and sharing experiences, knowledge, and ideas.
• Robert Kaplinsky shares math strategies, lessons, and resources designed to create problem solvers. The lessons are detailed and challenging!
• Mathigon “The mathematical playground” offers free manipulatives, activities, and lessons to make online learning interactive and engaging. The digital manipulates are a must-use!
• Fractal Foundation uses fractals to inspire interest in science, math and art. It has numerous fractal activities, software to help your students create their own fractals, and more.
• Greg Fletcher 3 Act Tasks contain engaging math videos with guiding questions. You can also download recording sheets to go with each video.

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

You might also like…

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

©2023 Teaching With a Mountain View . All Rights Reserved | Designed by Ashley Hughes

Username or Email Address

Remember Me

Lost your password?

Review Cart

No products in the cart.

Ways of thinking in STEM-based problem solving

• Original Paper
• Open access
• Published: 03 March 2023
• Volume 55 , pages 1219–1230, ( 2023 )

Cite this article

You have full access to this open access article

• Lyn D. English   ORCID: orcid.org/0000-0001-9118-2812 1  

8037 Accesses

10 Citations

Explore all metrics

This article proposes an interconnected framework, Ways of thinking in STEM-based Problem Solving , which addresses cognitive processes that facilitate learning, problem solving, and interdisciplinary concept development. The framework comprises critical thinking, incorporating critical mathematical modelling and philosophical inquiry, systems thinking, and design-based thinking, which collectively contribute to adaptive and innovative thinking. It is argued that the pinnacle of this framework is learning innovation, involving the generation of powerful disciplinary knowledge and thinking processes that can be applied to subsequent problem challenges. Consideration is first given to STEM-based problem solving with a focus on mathematics. Mathematical and STEM-based problems are viewed here as goal-directed, multifaceted experiences that (1) demand core, facilitative ways of thinking, (2) require the development of productive and adaptive ways to navigate complexity, (3) enable multiple approaches and practices, (4) recruit interdisciplinary solution processes, and (5) facilitate the growth of learning innovation. The nature, role, and contributions of each way of thinking in STEM-based problem solving and learning are then explored, with their interactions highlighted. Examples from classroom-based research are presented, together with teaching implications.

Similar content being viewed by others

Facilitating STEM Integration Through Design

Building a Competency-Based STEM Curriculum in Non-STEM Disciplines: A sySTEMic Approach

The Nature of Interdisciplinary STEM Education

Explore related subjects.

• Artificial Intelligence

Avoid common mistakes on your manuscript.

1 Introduction

With the prevailing emphasis on integrated STEM education, the power of mathematical problem solving has been downplayed. Over two decades we have witnessed a decline in research on mathematical problem solving and thinking, with more questions than answers emerging (English & Gainsburg, 2016 ; Lester & Cai, 2016 ). This is of major concern, especially since work and non-work life increasingly call for resources beyond “textbook” problem solving (Chin et al., 2019 ; Krause et al., 2021 ). Such changing demands could not have been more starkly exposed than in the recent COVID 19 crisis, where mathematics played a crucial role in public and personal discourse, in describing and modelling current and potential scenarios, and in explaining and justifying societal regulations and restrictions. As Krause et al. ( 2021 ) highlighted, “No mathematical task we can create could be a richer application of mathematics than this real situation” (p. 88).

Modelling and statistical analyses that led the search for strategies to “flatten the curve” with minimal social or economic detriment were prominent in the media (Rhodes & Lancaster, 2020 ; Rhodes et al., 2020 ). As nations strive to rebuild their economies including grappling with crippling energy costs, advanced modelling again plays a key role (Aviso et al., 2022 ; Oxford Economics, 2022 ; Teng et al., 2022 ). Unfortunately, the gap between the mathematical modelling applied during the pandemic and how the public interpreted the models has been “palpable and evident” (Aguilar & Castaneda, 2021 ). Sadly, as Di Martino (Krause et al., 2021 ) pointed out regarding his nation:

People’s reactions in this pandemic underlined the spread of a widely negative attitude toward mathematics among the adult population in Italy. We have witnessed the proliferation of strange, unscientific, and dangerous theories but also the risk of refusing to approach facts that involve math and the resulting dependence to fully rely on others when mathematics is used to justify decisions. Also, on this occasion many people showed their own fear of math and their rejection of mathematical arguments as relevant factors in justification (p. 93).

This heightened visibility of mathematics in society coupled with a general lack of mathematical literacy sparked major questions about the repercussions for education and research (Bakker & Wagner, 2020 ; Kollosche & Meyerhöfer, 2021 ). One such repercussion is the need to reconsider perspectives on STEM-based problem solving and how different ways of thinking can enhance or hinder solutions. With reference to engineering education, Dalal et al. ( 2021 ) indicated how numerous calls for a focus on ways of thinking have largely been taken for granted or at least treated at a superficial level. As these authors pointed out, while perceived as a theoretical concept, ways of thinking “can and should be used in practice as a structure for solution-oriented outlooks and innovation” (p. 109).

Given the above points, I propose an interconnected framework, Ways of thinking in STEM-based Problem Solving (Fig.  1 ), which addresses cognitive processes that facilitate learning, problem solving, decision-making, and interdisciplinary concept development (cf. Slavit et al., 2021 ). The framework comprises critical thinking (including critical mathematical modelling and philosophical inquiry), systems thinking, and design-based thinking. Collectively, these thinking skills contribute to adaptive and innovative thinking (McKenna, 2014 ) and ultimately lead to the development of learning innovation (Sect.  3.4 ; English, 2018 ).

These thinking skills have been chosen because of their potential to enhance STEM-based problem solving and interdisciplinary concept development (English et al., 2020 ; Park et al., 2018 ; Slavit et al., 2021 ). Highlighting these ways of thinking, however, is not denying the importance of other thinking skills such as creativity, which is incorporated within the adaptive and innovative thinking component of the proposed framework (Fig.  1 ), and is considered multidimensional in nature (OECD, 2022 ). Other key skills such as communication and collaboration (Stehle & Peters-Burton, 2019 ) are acknowledged but not explored here.

In line with Dalal et al. ( 2021 ), I consider the proposed ways of thinking as providing an organisational structure both individually and interactively when enacted in practice—notwithstanding the contextual and instructional influences that can have an impact here (Slavit et al., 2022 ). Prior to exploring these ideas, I consider STEM-based problem solving with a focus on mathematics.

2 STEM-based problem solving: a focus on mathematics

Within our STEM-intensive society, we face significant challenges in promoting STEM education from the earliest grades while also maintaining the integrity of the individual disciplines (Tytler, 2016 ). With the increasing need for STEM skills across multiple workforce domains, contrasted with difficulties in STEM implementation in many schools (e.g., Dong et al., 2020 ), the urgency to advance STEM education has never been greater. With the massive disruption caused by COVID-19, coupled with problematic international relations, our school students’ futures have become even more uncertain—we cannot ignore the rapid changes that will continue to impact their lives. Unlike business and industry, where disruption creates a “force-to-innovate” approach (Crittenden, 2017 , p. 14), much of school education seems oblivious to preparing students for these disruptive forces or at least are restricted in doing so by set curricula.

Preparing our students for an increasingly uncertain and complex future requires rethinking the nature of their learning experiences, in particular, the need for more relevant and innovative problems that are challenging but manageable, and importantly, facilitate adaptive learning and problem solving (McKenna, 2014 ). A failure to provide such opportunities may have detrimental effects on young students’ learning and their future achievements (Engel et al., 2016 ). Despite mathematics being cited as the core of STEM education and foundational to the other disciplines (e.g., Larson, 2017 ; Roberts et al., 2022 ; Shaughnessy, 2013 ), it is frequently ignored in integrated STEM activities (English, 2016 ; Maass et al., 2019 ; Mayes, 2019 ; Shaughnessy, 2013 ). For example, quantitative reasoning, which is critical to integrated STEM problem solving, is frequently “misrepresented, underdeveloped, and ignored in STEM classrooms” (Mayes, 2019, p. 113). Likewise, Tytler ( 2016 ) warned that there needs to be an explicit focus on the mathematical concepts and thinking processes that arise in STEM activities. Without this focus, STEM programs run the risk of reducing the valuable contributions of mathematical thinking. If children fail to see meaningful links between their learning in mathematics and the other STEM domains, they can lose interest not only in mathematics but also in the other disciplines (Kelley & Knowles, 2016 ).

Traditional notions of mathematical problem solving (e.g., Charles, 1985 ) are now quite inadequate when applied to our current world. At a time of increasing creative disruption, it is essential for mathematical problem solvers to be adaptive in dealing with unforeseen local, national, and international problems. Increasingly, STEM-based problems in the real world encompass more than just disciplinary content and practices. While not denying the essential nature of these components, issues pertaining to cultural, social, political, and ethical dimensions (Kollosche & Meyerhöfer, 2021 ; Pheasants, 2020) can also impact the solution process, necessitating the application of appropriate thinking skills. As Pheasants stressed, “If STEM education is to prepare students to grapple with complex problems in the real world, then more attention ought to be given to approaches that are inclusive of the non-STEM dimensions that exist in those problems.”

In light of the above arguments, I view mathematical and STEM-based problems as goal-directed experiences that (1) demand STEM-relevant ways of thinking, (2) require the development of productive and adaptive ways to navigate complexity, (3) enable multiple approaches and practices (Roberts et al., 2022 ), (4) recruit interdisciplinary solution processes, and (5) facilitate growth of learning innovation for all students regardless of their background (English, 2018 ). In contrast to traditional expectations, such problems need to embody affordances that facilitate learning innovation, where all students can move beyond their existing competence in standard problem solving and be challenged to generate new knowledge in solving unanticipated problems. Even students who achieve average results on standardised tests display conceptual understanding and advanced mathematical thinking not normally seen in the classroom—especially when current common practices emphasise number skills at the expense of problem solving and reasoning with numbers (Kazemi, 2020 ).

Limited attention, however, has been paid to how problem experiences can be developed that press beyond basic content knowledge (Anderson, 2014; Li, Schoenfeld et al., 2019), encompass the STEM disciplines, and develop important ways of thinking. In their recent article, Slavit et al. ( 2021 ) argued that STEM education should be “grounded in our knowledge of how students think in STEM-focused learning environments” (p. 1), and that fostering twenty-first-century skills is essential. Yet, as these authors highlighted, there is not much research on STEM ways of thinking, with even fewer theoretical perspectives and frameworks on which to draw. In the next section, I consider these ways of thinking, defined in Table 1 , and provide examples of their applications to STEM-based problem solving.

3 Ways of thinking in STEM-based problem solving

3.1 critical thinking.

Although long recognised as a significant process in a range of fields, research on critical thinking in education, especially in primary education, has been limited (Aktoprak & Hursen, 2022 ). Critical thinking has long been associated with mathematical reasoning and problem solving, but their association remains under-theorized (Jablonka, 2020 ). Likewise, connections between critical thinking and design thinking have had limited attention largely due to their shared conceptual structures not being articulated (Ericson, 2022 ). As a twenty-first century skill, critical thinking is increasingly recognised as essential in STEM and mathematics education (Kollosche & Meyerhöfer, 2021 ) but is sadly lacking in many school curricula (Braund, 2021 ). As applied to STEM-based problem solving, critical thinking builds on inquiry skills (Nichols et al., 2019 ) and entails evaluating and judging problem situations including statements, claims, and propositions made, analysing arguments, inferring, and reflecting on solution approaches and conclusions drawn. Although critical thinking can contribute significantly to each of the other ways of thinking, its application is often neglected. For example, critical thinking is increasingly needed in design and design thinking, which play a key role in product development, environmental projects, and even in forms of social interaction (as discussed in Sect.  3.3 ; Ericson, 2022 ).

3.1.1 Critical mathematical modelling

One rich source of problem experiences that foster critical thinking is that of modelling. The diverse field of mathematical modelling has long been prominent in the secondary years (e.g., Ärlebäck & Doerr, 2018 ) but remains under-researched in the primary years, especially in relation to its everyday applications. Effective engagement with social, political, and environmental issues through modelling and statistics demands critical thinking, yet such aspects are not often considered in school curricula (Jablonka, 2020 ). This is of particular concern, given the pressing need to tackle such issues in today’s world.

As noted in several publications, the interdisciplinary nature of mathematical modelling makes it ideal for STEM-based problem solving (English, 2016 ; Maass et al., 2019 ; Zawojewski et al., 2008 ). Numerous definitions of models and modelling exist in the literature (e.g., Blum & Leiss, 2007 ; Brady et al., 2015 ). For this article, modelling involves developing conceptual innovations in response to real-world needs; effective modelling requires moving beyond the conventional ways of thinking applied in typical school problems (Lesh et al., 2013 ) to include contextual and critical analysis.

Much has been written about the role of modelling during COVID-19. Mathematical models played a major role in grappling with COVID 19, but their projections were a source of controversy (Rhodes & Lancaster, 2020 ). There seems little appreciation of the critical nature of mathematical models in society (Barbosa, 2006 ) and how assumptions in the modelling process can sway decisions. STEM-based problem solving needs to incorporate not just modelling itself, but also critical mathematical modelling. Critiquing what a model yields, and what is learned, is of increasing social importance (Aguilar & Castaneda, 2021 ; Barbosa, 2006 ). Indeed, as Braund ( 2021 ) illustrated, the Covid-19 pandemic has revealed the urgent need for “critical STEM literacy” (p. 339)—an awareness of the complex and problematic interactions of STEM and politics, and a knowledge and understanding of the underlying STEM concepts and representations is essential:

There are two imperatives that emerge: first, that there is sufficient STEM literacy to negotiate the complex COVID-19 information landscape to enable personal decision taking and second, that this is accompanied by a degree of criticality so that politicians and experts are called to account (Braund, 2021 , p. 339).

Kollosche ( 2021 ) shed further light on the lack of critical thinking in the media’s reporting on COVID, with his argument that “most newspaper reports were effective in creating the problems” because of their focus on ideal forms of mathematical concepts and modelling, without discussing the assumptions and methods behind the reported data. Of concern is that “mass media still fail to present scientific models and results in a way that allows for mathematical reflection and a critical evaluation of such information by citizens.”

Modelling experiences that draw on students’ cultural and community contexts (Anhalt et al., 2018 ; English, 2021a , 2021b ; Turner et al., 2009 ) provide rich opportunities for critical thinking from a citizen’s perspective. Such opportunities can also assist students in appreciating that mathematics is not merely a means of calculating answers but is also a vehicle for social justice, where critical thinking plays a key role (Cirillo et al., 2016 ; Greer et al., 2007 ). In their studies of critical thinking in cultural and community contexts, Turner and her colleagues (Turner et al., 2009 , 2021 , 2022 ) explored culturally responsive, community-based approaches to mathematical modelling with elementary teachers and students. Using a range of authentic modelling contexts, Turner and her colleagues illustrated how students’ modelling processes generated a number of issues that required them to think critically about their lives and their lives within their community. In one activity, students were applying design thinking and processes as they redesigned their local park. They generated, for example, mathematical models to estimate how long children would have to wait to use the swings—this informed their decision that the park did not meet the needs of the community. This community-based modelling highlighted “ongoing negotiation between students’ experiences and intentions related to the community park, the constraints of the actual context, and the mathematical issues that arose” (Turner et al., 2009 , p. 148).

Another example of modelling involving critical thinking in cultural and community contexts was implemented in a sixth-grade Cyprus classroom. Students were required to develop a model for their country to purchase water supplies from a choice of nearby nations (English & Mousoulides, 2011 ). Students were to consider travel distances, water price, available supply per week, oil tanker capacity, costs of water and oil, and quality of the port facilities in the neighbouring countries. The targeted model had to select the best option not only for the present but also for the future. Students’ models ranged from a basic form, where port facilities and water supply were ignored, to more sophisticated models, where all factors were integrated, with carbon emissions also considered. One of the student groups who took into account environmental factors commented, “It would be better for the country to spend a little more money and reduce oil consumption. And there are other environmental issues, like pollution of the Mediterranean Sea.” The more sophisticated models reflected systems thinking (Sect.  3.2 ), where the impact of partial factors such as oil consumption on the whole domain (ocean ecosystems and community) was also considered.

3.1.2 Philosophical Inquiry

One underrepresented means of fostering critical thinking in mathematical and STEM-based problem solving is through philosophical inquiry (Calvert et al., 2017 ; English, 2013 , 2022 ; Kennedy, 2012 ; Mukhopadhyay & Greer, 2007 ). Such inquiry encourages a range of thinking skills in identifying hidden assumptions, determining alternative courses of action, and reflecting on conclusions drawn and claims made. Several studies have shown how engaging children in communities of philosophical inquiry nurtures critical thinking dispositions, which become both a goal and a method (Bezençon, 2020 ; Daniel et al., 2017 ; Lipman, 2003 , 2008 ). At the same time, philosophical inquiry can lead to “conceptual deepening” (Bezençon, 2017), where analysis of mathematical and related STEM concepts as they apply beyond the classroom can be fostered. Given the increased societal awareness of mathematics and STEM in recent years, philosophical inquiry can be a powerful tool in enhancing students’ understanding and appreciation of how these disciplines shape societies. At the same time, philosophical inquiry can stimulate consideration of ethical issues in the applications of these disciplines (Bezençon, 2020 ). For example, Mukhopadhyay and Greer ( 2007 ) indicated how mathematics education should “convey the complexity of mathematical modeling social phenomena and a sense of what demarcates questions that can be answered by empirical evidence from those that depend on value systems and world-views” (p. 186).

A comprehensive review by O’Reilly et al. ( 2022 ) identified pedagogical approaches to scaffolding early critical thinking skills including inquiry-based teaching using classroom dialogue or questioning techniques. Such techniques include philosophical inquiry and encouraging children to construct, share, and justify their ideas regarding a task or investigation. Other opportunities for philosophical inquiry include group problem solving, peer sharing of created models, and facilitating critical and constructive peer feedback. For example, Gallagher and Jones ( 2021 ) reported on integrating mathematical modelling and economics, where beginning teachers were presented with a task involving a problematic community issue following a school shooting. In such cases, numerous courses of action are typically proposed for addressing the problem. Not surprisingly, various community opinions exist on such proposals, giving rise to valuable contexts for philosophical inquiry and critical modelling, where data and their sources are carefully analysed. With the escalation of statistical data from the mass media, it is imperative to commence the foundations of critical and philosophical thinking early. Students’ skills in asking critical questions as they work with data in constructing and improving a model, reflect on what their models convey, consider consequences of their models, and justify and communicate their conclusions require nurturing throughout school (Gibbs & Young Park, 2022 ).

3.2 Systems thinking

Systems thinking cuts across the STEM disciplines as well as many other fields outside education. It is considered a key component of “critical thinking and problem solving” in 21st Century Learning (P21, 2015 ) and is often cited as a “habit of mind” in engineering education (e.g., Lippard et al., 2018 ; Lucas et al., 2014 ). Numerous definitions exist for systems thinking (e.g., Bielik et al., 2022 ; Damelin et al., 2017 ; Jacobson & Wilenski, 2022 ), with Bielik et al. ( 2022 ) identifying such thinking as the ability to “consider the system boundaries, the components of the system, the interactions between system components and between different subsystems, and emergent properties and behaviour of the system” (p. 219). In more basic terms, Shin et al. ( 2022 ) refer to systems thinking as “the ability to understand a problem or phenomenon as a system of interacting elements that produces emergent behavior” (p. 936).

Systems thinking interacts with the other thinking forms including those displayed in Fig.  1 , as well as computational thinking (Shin et al., 2022), critical thinking (Curwin et al., 2018 ) and mathematical thinking more broadly (Baioa & Carreira, 2022 ). Systems thinking is considered especially important in conceptualizing a problematic situation within a larger context and in perceiving problems in new and different ways (Stroh, 2018 ). Of special relevance to today’s world is the realisation that perfect solutions do not exist and the choices one makes in applying systems thinking will impact on other parts of the system (Meadows, 2008 ). What is often not considered in today’s complex societies—at least not to the extent required—is that we live in a world of intrinsically linked systems, where disruption in one part will reverberate in others. We see so many instances where particular courses of action are advocated or mandated in societal systems, while the impact on sub-components is perilously ignored. Examples are evident in many nations’ responses to COVID-19, where escalating lockdowns impacted economies and communities, whose demands had to be balanced against purely epidemiological factors. The reverberations of such actions stretch far and wide over long periods. Likewise, the various impacts of current climate actions are frequently ignored, such as how the construction of vast areas of renewable resources (e.g., wind turbines) can have deleterious effects on the surrounding environments including wildlife.

Despite its centrality across the STEM domains, systems thinking is almost absent from mathematics education (Curwin et al., 2018 ). This is despite claims by many researchers that modelling, systems thinking, and associated thinking processes should be significant components of students’ education (Bielik et al., 2022 ; Jacobson & Wilenski, 2022 ). Indeed, systems thinking is featured prominently in the US A Framework for K-12 Science Education (NRC, 2012 ) and the Next Generation Science Standards (NGSS Lead States, 2013 ), and has received considerable attention in science education (e.g., Borge, 2016 ; Hmelo-Silver et al., 2017 ; York et al., 2019 ) and engineering education (e.g., Lippard & Riley, 2018 ; Litzinger, 2016 ).

Given the complexity of systems thinking in today’s world, and the diverse ways in which it is applied, students require opportunities to experience how systems thinking can interact with other forms such as design thinking and critical thinking (Curwin et al., 2018 ; Shin et al., 2022). Such interactions occur in many popular STEM investigations including one in which fifth-grade students designed, constructed, and experimented with a loaded paper plane (paper clips added) in determining how load impacts on the distance travelled (English, 2021b ). Students observed that changing one design feature (e.g., wingspan or load position) unsettles some other features (e.g., fuselage depth decreases wingspan). The investigation yielded an appreciation of the intricacy of interactions among a system’s components, and their scarcely predictable mutual effects.

Other studies have revealed how very young children can engage in basic systems thinking within STEM contexts. For example, Feriver et al. ( 2019 ) administered a story reading session to individual 4- to 6-year-old children in preschools in Turkey and Germany, and then followed this with individual semi-structured interviews about the story. The reading session was based on the story book, “The Water Hole” (Base, 2001 ), which draws on basic concepts of systems within an ecosystem context. Their study found the young children to have some complex understanding of systems thinking in terms of detecting obvious gradual changes and two-step domino and/or multiple one-way causalities, as well as describing the behaviour of a balancing loop (corrective actions that try to reduce the gap between a desired level or goal and the actual lever, such as temperature and plant growth). However, the children understandably experienced difficulties in several areas, in which even adults have problems. For example, detecting a reinforcing loop, identifying unintended consequences, detecting hidden components and processes, adopting a multi-dimensional perspective, and predicting how a system would behave in the future were problematic for them. In another study, Gillmeister ( 2017 ) showed how preschool children have a more complex understanding of systems thinking than previously claimed. Their ability to utilize simple systems thinking tools, such as stock-flow maps, feedback loops and behaviour over time graphs, was evident.

Although the research has not been extensive, current findings indicate how we might capitalise on the seeds of early systems thinking across the STEM fields. With the pervasive nature of systems thinking, it is argued that its connections to other forms of thinking across STEM should be nurtured (cf., Svensson, 2022 ). This includes links to design-based thinking where designed products, for example, operate “within broader systems and systems of systems” (Buchanan, 2019 ).

3.3 Design-based thinking and STEM-based problem solving

Design-based thinking plays a major role in complex problem solving, yet its contribution to mathematics learning has been largely ignored, especially at the primary levels. Design is central in technology and engineering practice (English et al., 2020 ; Guzey et al., 2019 ), from bridge construction through to the development of medical tools. Design contributes to all phases of problem solving and drives students toward innovative rather than predetermined outcomes (Goldman & Zielezinski, 2016 ). As applied to STEM education, design thinking is generally viewed as a set of iterative processes of understanding a problem and developing an appropriate solution. It includes problem scoping, idea generation, systems thinking, designing and creating, testing and reflecting, redesigning and recreating, and communicating. Both learning about design and learning through design can help students develop more informed analytical approaches to mathematical and STEM-based problem solving (English et al., 2020 ; Strimel et al., 2018 ).

Learning about design in solving complex problems emphasises design thinking processes per se, that is, students learn about design itself. Design thinking as a means for learning about design is not as prevalent in integrated STEM education. Although learning through and about design should not be divorced during problem solving (van Breukelen et al., 2017 ), one learning goal can take precedence over the other. Both require greater attention, especially in today’s design-focused world (Tornroth & Wikberg Nilsson, 2022 ).

A plurality of solutions of varying levels of sophistication is possible in applying design thinking processes, so students from different achievement levels can devise solutions (Goldman & Kabayadondo, 2017 )—and research suggests that lower-achieving students benefit most (Chin et al., 2019 ). The question of how best to develop design learning across grades K-12, however, has not received adequate attention. This is perhaps not surprising given that comparatively few studies have investigated the design thinking of grades 6–12 and even fewer across the elementary years (Kelley & Sung, 2017 ; Strimel et al., 2018 ).

Also overlooked is how design-based thinking can foster concept development, that is, how it can foster “learning while designing”, or generative learning (English et al., 2020 ). Design-based thinking provides natural opportunities to develop understanding of the required STEM concepts (Hjalmarson et al., 2020 ). As these authors pointed out, when students create designs, they represent models of their thinking. When students have to express their conceptual understanding in the form of design, such as an engineering designed product, their knowledge (e.g., of different material properties) is tested. Studies have revealed such learning occurs when elementary and middle school students design and construct various physical artifacts. For example, in a fifth-grade study involving designing and building an optical instrument that enabled one to see around corners, King and English ( 2017 ) observed how students applied and enriched their understanding of scientific concepts of light and how it travels through a system, together with their mathematical knowledge of geometric properties, angles and measurement.

In a study in the secondary grades (Langman et al. ( 2019 ), student groups completed modules that included a model-eliciting activity (Lesh & Zawojewski, 2007 ) involving iterative design thinking processes. Students were to interpret, assess, and compare images of blood vessel networks grown in scaffolds, develop a procedure or tool for measuring (or scoring) this vessel growth, and demonstrate how to apply their procedure to the images, as well as to any image of a blood vessel network. The results showed how students designed a range of mathematical models of varying levels of sophistication to evaluate the quality of blood vessel networks and developed a knowledge of angiogenesis in doing so.

In another study implemented in a 6th-grade classroom, students were involved in both learning about and learning through design in creating a new sustainable town with the goal of at least 50% renewable energy sources. The students were also engaged in systems thinking, as well as critical thinking, as they planned, developed, and critically analysed the layout and interactions of different components of their town system (e.g., where to place renewable energy sources to minimise impact on the residences and recreational areas; where to situate essential services to reach different town components). Students also had to consider the budgetary constraints in their town designs. Their learning about design was apparent in their iterative processes of problem scoping, idea generation and modification, balancing of benefits and trade-offs (dealing with system subcomponents), critical reflections on their designs, and improvements of the overall town design. Learning though design was evident as students displayed content knowledge pertaining to renewable energy and community needs, applied spatial reasoning in positioning town features, and considered budgetary constraints in reaching their “best” design.

Engineering design processes form a significant component of design thinking and learning, with foundational links across the STEM disciplines including mathematical modelling. Engineering design-based problems help students appreciate how they can apply different ideas and approaches from the STEM disciplines, with more than one outcome possible (Guzey et al., 2019 ). Such design thinking engages students in interpreting problems, developing initial design ideas, selecting and testing a promising design, analysing results from their prototype solution, and revising to improve outcomes.

It is of concern that limited attention has been devoted to engineering design-based thinking in mathematics and the other STEM disciplines in the primary grades. This is despite research indicating how very young children can engage in design-based thinking when provided with appropriate opportunities (e.g., Cunningham, 2018 ; Elkin et al., 2018 ). Elkin et al., for example, used robotics in early childhood classrooms to introduce foundational engineering design thinking processes. Their study illustrated how these young learners used engineering design journals and engaged in design thinking as they engineered creative solutions to challenging problems presented to them. It seems somewhat paradoxical that foundational engineering design thinking is a natural part of young children’s inquiry about their world, yet this important component has been largely ignored in these informative years.

In sum, design-based thinking is a powerful and integrative tool for mathematical and STEM-based problem solving and requires greater focus in school curricula. Both learning about and through design have the potential to improve learning and problem solving well beyond the school years (Chin et al., 2019 ). At the same time, design-based thinking, together with the other ways of thinking, can contribute significantly to learning innovation (English, 2018 ).

3.4 Adaptive and innovative thinking for learning innovation

Disruption is rapidly becoming the norm in almost all spheres of life. The recent national and international upheavals have further stimulated disruptive thinking (or disruptive innovation), where perspectives on commonly accepted (and often inefficient) solutions to problems are rejected for more innovative approaches and products. Given the pressing need for problem solvers capable of developing new and adaptable knowledge, rather than applying limited simplistic procedures or strategies, we need to foster what I term, learning innovation (English, 2018 ; Fig.  2 ). Such learning builds on core content to generate more powerful disciplinary knowledge and thinking processes that can, in turn, be adapted and applied to solving subsequent challenging problems.

Learning innovation

Figure  2 (adapted from McKenna, 2014 ) illustrates how learning innovation can be fostered through the growth of mathematical and STEM knowledge, together with STEM ways of thinking. The optimal adaptability corridor represents the growth of mathematical and STEM-based problem solving from novice to adaptable solver . Adaptable problem solvers (Hatano & Oura, 2003 ) have developed the flexibility, curiosity, and creativity to tackle novel problems—skills that are needed in jobs of the future (Denning & Noray, 2020 ; OECD, 2019 ). Without the disciplinary knowledge, ways of thinking, and engagement in challenging but approachable problems, a student risks remaining merely a routine problem solver—one who is skilled solely in applying previously taught procedures to solve sets of familiar well-worn problems, as typically encountered at school. Learning innovation remains a challenging and unresolved issue across curriculum domains and is proposed as central to dealing effectively with disruption.

Particularly rich experiences for developing learning innovation involve interdisciplinary modelling, incorporating critical mathematical modelling (e.g., Hallström and Schönborn, 2019 ; Lesh et al., 2013 ). As noted, a key feature of model-eliciting experiences is the affordances they provide all students to exhibit “extraordinary abilities to remember and transfer their tools to new situations” (Lesh et al., 2013 , p. 54). Modelling enables students to apply more sophisticated STEM concepts and generate solutions that extend beyond their usual classroom problems. Such experiences require different ways of thinking in problem solving as they deal with, for example, conflicting constraints and trade-offs, alternative paths to follow, and various tools and representations to utilise. In essence, interdisciplinary modelling may be regarded as a way of creating STEM concepts, with modeling and concept development being highly interdependent and mutually supportive (cf., Lesh & Caylor, 2007 ).

4 Concluding points

The ill-defined problems of today, coupled with unexpected disruptions across all walks of life, demand advanced problem-solving by all citizens. The need to update outmoded forms of problem solving, which fail to take into account increasing global challenges, has never been greater (Cowin, 2021 ). The ways-of-thinking framework has been proposed as a powerful means of enhancing problem-solving skills for dealing with today’s unprecedented game-changers. Specifically, critical thinking (including critical mathematical modelling and philosophical inquiry), systems thinking, and design-based thinking are advanced as collectively contributing to the adaptive and innovative skills required for problem success. It is argued that the pinnacle of this framework is learning innovation, which can be within reach of all students. Fostering students’ agency for developing learning innovation is paramount if they are to take some control of their own problem solving and learning (English, 2018 ; Gadanidis et al., 2016 ; Roberts et al., 2022 ).

Establishing a culture of empowerment and equity, with an asset-based approach, where the strengths of all students are recognised, can empower students as learners and achievers in an increasingly uncertain world (Celedón-Pattichis et al., 2018 ). Teacher actions that encourage students to express their ideas, together with a program of future-oriented mathematical and STEM-based problems, can nurture students’ problem-solving confidence and dispositions (Goldman & Zielezinski, 2016 ; Roberts et al., 2022 ). In particular, mathematical and STEM-based modelling has been advocated as a rich means of developing multiple ways of thinking that foster adaptive and innovative learners—learners with a propensity for developing new knowledge and skills, together with a willingness to tackle ill-defined problems of today and the future. Such propensity for dealing effectively with STEM-based problem solving is imperative, beginning in the earliest grades. The skills gained can thus be readily transferred across disciplines, and subsequently across career opportunities.

Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Aguilar, M. S., & Castaneda, A. (2021). What mathematical competencies does a citizen need to interpret Mexico’s official information about the COVID-19 pandemic? Educational Studies in Mathematics . https://doi.org/10.1007/s10649-021-10082-9

Article   Google Scholar  

Aktoprak, A., & Hursen, C. (2022). A bibliometric and content analysis of critical thinking in primary education. Thinking Skills and Creativity, 44 , 101029.

Anhalt, C. O., Staats, S., Cortez, R., & Civil, M. (2018). Mathematical modeling and culturally relevant pedagogy. In Y. J. Dori, Z. R. Mevarech, & D. R. Baker (Eds.), Cognition, metacognition, and culture in STEM education (pp. 307–330). Springer. https://doi.org/10.1007/978-3-319-66659-4_14

Chapter   Google Scholar  

Ärlebäck, J. B., & Doerr, H. M. (2018). Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence. ZDM Mathematics Education, 50 (1–2), 187–200.

Aviso, K. B., Yu, C. D., Lee, J.-Y., & Tan, R. R. (2022). P-graph optimization of energy crisis response in Leontief systems with partial substitution. Cleaner Energy and Technology, 9 , 100510. https://doi.org/10.1016/j.clet.2022.100510

Baioa, A. M., & Carreira, S. (2022). Mathematical thinking about systems—Students modeling a biometrics identity verification system. Mathematical Thinking and Learning.

Bakker, A., & Wagner, D. (2020). Pandemic: Lessons for today and tomorrow? Educational Studies in Mathematics, 104 , 1–4. https://doi.org/10.1007/s10649-020-09946-3

Barbosa, J. C. (2006). Mathematical modelling in classroom: A socio-critical and discursive perspective. ZDM Mathematics Education, 38 , 293–301. https://doi.org/10.1007/BF02652812

Base, G. (2001). In N. Harry (Ed.), The Water Hole. Abrs Inc.

Google Scholar  

Benessiaa, A., & De Marchi, B. (2017). When the earth shakes … and science with it. The management and communication of uncertainty in the L’Aquila earthquake. Futures, 91 , 35–45.

Bezençon, E. (2020). Practising philosophy of mathematics with children: what, why and how? Philosophy of Mathematics Education Journal, 36 .

Bielik, T., Stephens, L., McIntyre, C., Damelin, D., & Krajcik, J. S. (2022). Supporting student system modelling practice through curriculum and technology design. Journal of Science Education and Technology, 31 , 217–231. https://doi.org/10.1007/s10956-021-09943-y

Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12)—Education, engineering and economics. Horwood.

Borge, M. (2016). Systems thinking as a design problem. In R. A. Duschl & A. S. Bismack (Eds.), Reconceptualizing STEM education (pp. 68–80). Routledge.

Brady, C., Lesh, R., & Sevis, S., et al. (2015). Extending the reach of the models and modelling perspective: A course-sized research site. In G. A. Stillman (Ed.), Mathematical modelling in education research and practice (pp. 55–66). Springer.

Braund, M. (2021). Critical STEM Literacy and the COVID-19 Pandemic. Canadian Journal of Science, Mathematics, and Technology Education, 21 , 39–356. https://doi.org/10.1007/s42330-021-00150-w

Buchanan, R. (2019). Systems thinking and design thinking: the search for principles in the world we are making. She Ji the Journal of Design, Economics, and Innovation, 5 (2), 85–104.

Calvert, K., Forster, M., Hausberg, A., et al. (2017). Philosophizing with children in science and mathematics classes. In M. Rollins Gregory, J. Haynes, & K. Murris (Eds.), The Routledge international handbook of philosophy for children (pp. 191–199). Routledge.

Celedón-Pattichis, S., Lunney Borden, L., Pape, S. J., et al. (2018). Asset-based approaches to equitable mathematics education research and practice. Journal for Research in Mathematics Education, 49 (4), 373–389.

Charles, R. (1985). The role of problem solving. Arithmetic Teacher, 32 , 48–50.

Chin, D. B., Blair, K. P., Wolf, R. C., Conlin, L. D., Cutumisu, M., Pfaffman, J., & Schwartz, D. L. (2019). Educating and measuring choice: A test of the transfer of design thinking in problem solving and learning. Journal of the Learning Sciences, 28 (3), 337–380. https://doi.org/10.1080/10508406.2019.1570933

Cirillo, M., Bartell, T. G., & Wager, A. A. (2016). Teaching mathematics for social justice through mathematical modeling. In C. R. Hirsch & A. R. RothMcDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 87–96). National Council of Teachers of Mathematics.

Cowin, J. (2021). The fourth industrial revolution: Technology and education. Journal of Systemics, Cybernetics and Informatics, 19 (8), 53–63.

Crittenden, AB, Crittenden VI, Crittenden, WF (2017) Industry transformation via channel disruption. Journal of Marketing Channels , 24 (1–2):13–26.

Cunningham, C. M. (2018). Engineering in elementary STEM education: Curriculum design, instruction, learning, and assessment . Teachers College Press.

Curwin, M. S., Ardell, A., MacGillivray, L., & Lambert, R. (2018). Systems thinking in a second grade curriculum: Students engaged to address a statewide drought. Frontiers in Education . https://doi.org/10.3389/feduc.2018.00090

Dalal, M., Carberry, A., Archambault, L., et al. (2021). Developing a ways of thinking framework for engineering education research. Studies in Engineering Education, 1 (2), 108–125.

Damelin, D., Krajcik, J., Mcintyre, C., & Bielik, T. (2017). Students making systems models: An accessible approach. Science Scope, 40 (5), 78–82.

Daniel, M., Gagnon, M., & Auriac-Slusarczyk, E. (2017). Dialogical critical thinking in kindergarten and elementary school: Studies on the impact of philosophical praxis in pupils. In M. R. Gregory, J. Haynes, & K. Murris (Eds.), The Routledge international handbook of philosophy for children (pp. 236–244). Routledge.

Denning, D. J., & Noray, K. (2020). Earnings dynamics, changing job skills, and STEM careers. The Quarterly Journal of Economics, 135 (4), 1965–2005. https://doi.org/10.1093/qje/qjaa021

Dong, Y., Wang, J., Yang, Y., & Karup, P. M. (2020). Understanding intrinsic challenges to STEM instructional practices for Chinese teachers based on their beliefs and knowledge base. International Journal of STEM Education, 7 , 47. https://doi.org/10.1186/s40594-020-00245-0

Eleyyan, S. (2021). The future of education according to the fourth industrial revolution. Journal of Educational Technology and Online Learning, 4 (1), 23–30.

Elkin, M., Sullivan, A., & Bers, M. U. (2018). Books, butterflies, and ’bots: Integrating engineering and robotics into early childhood curricula. In L. D. English & T. Moore (Eds.), Early engineering learning (pp. 225–248). Springer.

Engel, M., Claessens, A., Watts, T., & Farkas, G. (2016). Mathematics content coverage andstudent learning in kindergarten. Educational Researcher , 45 (5), 293–300. https://doi.org/10.3102/0013189X16656841

English, L. D. (2013). Modeling as a vehicle for philosophical inquiry in the mathematics classroom. The Journal of Analytic Teaching and Philosophical Praxis, 34 (1), 46–57.

English, L. D. (2016). STEM education K-12: Perspectives on integration. International Journal of STEM Education, 3 (1), 1–12. https://doi.org/10.1186/s40594-016-0036-1

English, L. D. (2018). Disruption and learning innovation across STEM. Plenary presented at the 5th International Conference of STEM in Education , Brisbane. https://stem-in-ed2018.com.au/proceedings-2/

English, L. D. (2021a). Mathematical and interdisciplinary modeling in optimizing young children’s learning. In J. Suh, M. Wickstrom, & L. D. English (Eds.), Exploring mathematical modeling with young learners (pp. 3–24). Springer.

English, L. D. (2021b). Facilitating STEM integration through design. In J. Anderson & Y. Li (Eds.), Integrated approaches to STEM education: An international perspective (pp. 45–66). Springer.

English, L. D. (2022). Mathematical modeling and philosophical inquiry in the elementary school. In N. Kennedy (Ed.), Dialogical inquiry in mathematics teaching and learning: A philosophical approach. Lit Verlag.

English, L. D., Adams, R., & King, D. (2020). Learning by design across STEM education. In C. J. Johnson, M. M. Schroeder, T. Moore, & L. D. English (Eds.), Handbook of research on STEM education (pp. 76–96). Routledge.

English, L. D., & Gainsburg, J. (2016). Problem solving in a 21st-century mathematics curriculum. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 313–335). Taylor & Francis.

English, L. D., & Mousoulides, N. (2011). Engineering-based modelling experiences in the elementary classroom. In M. S. Khine & I. M. Saleh (Eds.), Models and modeling: Cognitive tools for scientific enquiry (pp. 173–194). Springer.

Ericson, J. D. (2022). Mapping the relationship between critical thinking and design thinking. Journal of the Knowledge Economy , 13 , 406–429.

Feriver, S., Olgan, R., Teksöz, G., & Barth, M. (2019). Systems thinking skills of preschool children in early childhood education contexts from Turkey and Germany. MDPI: Sustainability, 11 , 1478.

Gadanidis, G., Hughes, J. M., Minniti, L., & White, B. J. (2016). Computational thinking, Grade 1 students and the binomial theorem. Digital Experiences in Mathematics Education . https://doi.org/10.1007/s40751-016-0019-3

Gallagher, M. A., & Jones, J. P. (2021). Supporting students’ critical literacy: Mathematical modeling and economic decisions. In J. Suh, M. Wickstrom, & L. D. English (Eds.), Exploring mathematical modeling with young learners (pp. 373–388). Springer.

Gibbs, A. M., & Young Park, J. (2022). Unboxing mathematics: Creating a culture of modeling as critic. Educational Studies in Mathematics, 110 , 167–192. https://doi.org/10.1007/s10649-021-10119-z

Gillmeister, K. M. (2017). Development of early conceptions in systems thinking in an environmental context: An exploratory study of preschool students’ understanding of stocks and flows . State University of New York.

Goldman, S., & Kabayadondo, Z. (Eds.). (2017). Taking Design Thinking to School: How the technology of design can transform teachers, learners, and classrooms (pp. 3–19). Routledge.

Goldman, S., & Zielezinski, M. B. (2016). Teaching with design thinking: developing newvision and approaches to twenty-first century learning. In L. Annetta & J. Minogue (Eds.), Connecting science and engineering education practices in meaningful ways. Contemporary trends and issues in science education (p. 44). Springer. https://doi.org/10.1007/978-3-319-16399-4_10 .

Greer, B. (2009). Estimating Iraqi deaths: A case study with implications for mathematics education. ZDM Mathematics Education, 41 (1–2), 105–116. https://doi.org/10.1007/s11858-008-0147-3

Greer, B., Verschaffel, L., & Mukhopadhyay, S. (2007). Modeling for life: Mathematics and children’s experience. In W. Blum, W. Henne, & M. Niss (Eds.), Applications and modeling in mathematics education (ICMI study 14) (pp. 89–98). Kluwer.

Guzey, S. S., Ring-Whalen, E. A., Harwell, M., & Peralta, Y. (2019). Life STEM: A case study of life science learning through engineering design. International Journal of Science and Mathematics Education, 17 (1), 23–42.

Hallström, J., & Schönborn, K. J. (2019). Models and modelling for authentic STEM education: reinforcing the argument. International Journal of STEM Education , 6 , 22. https://doi.org/10.1186/s40594-019-0178-z .

Hatano, G., & Oura, Y. (2003). Commentary: Reconceptualizing school learning using insight from expertise research. Educational Researcher, 32 (8), 26–29.

Hjalmarson, M. A., Holincheck, N., Baker, C. K., & Galanti, T. M. (2020). Learning models and modeling across the STEM disciplines. In C. J. Johnson, M. M. Schroeder, T. Moore, & L. D. English (Eds.), Handbook of research on STEM education (pp. 223–233). Routledge.

Hmelo-Silver, C. E., Jordan, R., Eberbach, C., & Sinha, S. (2017). Systems learning with a conceptual representation: A quasi-experimental study. Instructional Science, 45 , 53–72. https://doi.org/10.1007/s11251-016-9392-y

Jablonka, E. (2020). Critical thinking in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_35

Jacobson, M. J., & Wilenski, U. (2022). Complex systems and the learning sciences: Educational, theoretical, and methodological implications. In K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 504–522). Cambridge University Press. https://doi.org/10.1017/9781108888295.031

Kazemi, E. (2021). Mathematical modeling with young learners: A commentary. In J. Suh, M. Wickstrom, & L. D. English (Eds.), Exploring mathematical modeming with young learners (pp. 337–342). Springer.

Kelley, T. R., & Sung, E. (2017). Examining elementary school students’ transfer of learning through engineering design using think-aloud protocol analysis. Journal of Technology Education, 28 (2), 83–108.

Kelley, T. R., & Knowles, J. G. (2016). A conceptual framework for integrated STEMeducation. International Journal of STEM Education , 3 (1), 1–11. https://doi.org/10.1186/s40594-016-0046-z .

Kennedy, N. S. (2012). What are you assuming? Mathematics Teaching in the Middle School, 18 (2), 86–91.

King, D. T., & English, L. D. (2017). Engineering design in the primary school: Applying STEM concepts to build an optical instrument. International Journal of Science Education, 38 , 2762–2794.

Koh, J. H. L., Chai, C. S., Wong, B., & Hong, H.-Y. (2015). Design thinking for education: Conceptions and applications in teaching and learning . Springer.

Book   Google Scholar  

Kollosche, D. (2021). Styles of reasoning for mathematics education. Educational Studies in Mathematics , 107 (3), 471–486.

Kollosche, D., & Meyerhöfer, W. (2021). COVID-19, mathematics education, and the evaluation of expert knowledge. Educational Studies in Mathematics, 108 , 401–417. https://doi.org/10.1007/s10649-021-10097-2 .

Krause, C. M., Di Martino, P., & Moschkovich, J. N. (2021). Tales from three countries: Reflections during COVID-19 for mathematical education in the future. Educational Studies in Mathematics, 108 , 87–104. https://doi.org/10.1007/s10649-021-10066-9 .

Langman, C., Zawojewski, J., McNicholas, P., Cinar, A., Brey, E., Bilgic, M., & Mehdizadeh, H. (2019). Disciplinary learning from an authentic engineering context. Journal of Pre-College Engineering Education Research, 9 , 77–94.

Larson M (2017) Math education is STEM education! NCTM president’s message . Retrieved from https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Math-Education-Is-STEM-Education .

Lesh, R., & Caylor, B. (2007). Introduction to the special issue: Modeling as application versus modeling as a way to create mathematics. International Journal of Computers for Mathematical Learning, 12 (3), 173–194.

Lesh, R., English, L. D., Riggs, C., & Sevis, S. (2013). Problem solving in the primary school (K-2). Mathematics Enthusiast, 10 (1), 35–60. https://doi.org/10.54870/1551-3440.1259 .

Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Information Age Publishing.

Lester, F., & Cai, F. (2016). Can mathematical problem solving be taught? In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems (pp. 117–136). Springer.

Li, Y, Schoenfeld A H (2019) Problematizing teaching and learning mathematics as ‘given’ in STEM education. International Journal of STEM Education, 6 , 44. https://doi.org/10.1186/s40594-019-0197-9 .

Lipman, M. (2003). Thinking in education (2nd ed.). Cambridge University Press.

Lipman, M. (2008). A life teaching thinking . The Institute for the Advancement of Philosophy for Children.

Lippard, C. N., Riley, K. L., & Mann, M. H. (2018). Encouraging the development of engineering habits of mind in prekindergarten learners. In L. D. English & T. J. Moore (Eds.), Early engineering learning (pp. 19–37). Springer.

Litzinger, T. A. (2016). Thinking about a system and systems thinking in engineering. In R. A. Duschl & A. S. Bismack (Eds.), Reconceptualizing STEM education (pp. 35–48). Routledge.

Lucas, B., Claxton, G., & Hanson, J. (2014). Thinking like an engineer: Implications for the education system. A report for the Royal Academy of Engineering Standing Committee for Education and Training.

Maass, K., Geiger, V., Romero Ariza, M., & Goos, M. (2019). The role of mathematics in interdisciplinary STEM education. ZDM Mathematics Education, 51 , 869–884. https://doi.org/10.1007/s11858-019-01100-5 .

Mayes, R. (2019). Quantitative reasoning and its role in interdisciplinarity. In B. Doig, J. Williams, D. Swanson, R. Borromeo Ferri, & P. Drake (Eds.), Interdisciplinary Mathematics Education. ICME-13 Monographs . Springer. https://doi.org/10.1007/978-3-030-11066-6_8 .

McKenna, A. F. (2014). Adaptive expertise and knowledge fluency in design and innovation. In A. Johri & B. M. Olds (Eds.), Cambridge handbook of engineering education research (pp. 227–242). Cambridge University Press.

Meadows, D. (2008). Thinking in systems: A primer (Chelsea Green Publishing). https://www.chelseagreen.com/product/thinking-in-systems .

Mukhopadhyay, S., & Greer, B. (2007). How many deaths? Education for statistical empathy. In B. Sriraman (Ed.), International perspectives on social justice in mathematics education (pp. 169–189). Information Age Publishing.

National Research Council (NRC). (2012). A framework for K-12 science education: Practices, crosscutting concepts, and core ideas . National Academies Press.

National Science and Technological Council. (2018). Charting a course for success: America’s strategy for STEM education. Committee on STEM Education of the National Science and Technological Council . https://www.whitehouse.gov/wp-content/uploads/2018/12/STEM-Education-Strategic-Plan-2018.pdf .

NGSS Lead States. (2013). Next generation science standards: For states, by states (The National Academic Press). http://www.nextgenscience.org/next-generation-science-standards .

Nichols, R. K., Burgh, G., & Fynes-Clinton, L. (2019). Reconstruction of thinking across the curriculum through the community of inquiry. In M. Rollins Gregory, J. Haynes, & K. Murris (Eds.), The Routledge international handbook of philosophy for children (pp. 245–252). Routledge.

OECD (2019). PISA 2022 Mathematics Framework . https://pisa2022-maths.oecd.org/ca/index.html .

O’Reilly, C., Devitt, A., & Hayes, N. (2022). Critical thinking in the preschool classroom—A systematic literature review. Thinking Skills and Creativity. https://doi.org/10.1016/j.tsc.2022.101110 .

Organisation for Economic Co-operation and Development (OECD). (2019). Education at a glance 2019: OECD indicators . OECD.

Organisation for Economic Co-operation and Development (OECD). (2022). Thinking Outside the box: The PISA 2022 Creative Thinking Assessment . OECD.

Oxford Economics (2022). Global economic model . https://www.oxfordeconomics.com/service/subscription-services/macro/global-economic-model/ .

P21 Partnership for 21st Century Learning (2015). http://www.p21.org/our-work/p21-framework .

Park, D.-Y., Park, M.-H., & Bates, A. B. (2018). Exploring young children’s understanding about the concept of volume through engineering design in a STEM activity: A case study. International Journal of Science and Mathematics Education, 16 (2), 275–294.

Rhodes, T., & Lancaster, K. (2020). Mathematical models as public troubles in COVID-19 infection control. Health Sociology Review, 29 , 177–194.

Rhodes, T., Lancaster, K., & Rosengarten, M. (2020). A model society: Maths, models and expertise in viral outbreaks. Critical Public Health, 30 (3), 253–256.

Roberts, T., Maiorca, C., Jackson, C., & Mohr-Schroeder, M. (2022). Integrated STEM as problem-solving practices. Investigations in Mathematics Learning . https://doi.org/10.1080/19477503.2021.2024721 .

Rollins Gregory, M., Haynes, J., & Murris, K. (Eds.). (2017). The Routledge international handbook of philosophy for children . Routledge.

Shaughnessy, M. (2013). By way of introduction: Mathematics in a STEM context. Mathematics Teaching in the Middle School , 18 (6):324.

Shin, N., Bowers, J., Roderick, S., McIntyre, C., Stephens, A. L., Eidin, E., et al. (2022). A framework for supporting systems thinking and computationalthinking through constructing models. Instructional Science , 50 (6), 933–960.

Slavit, D., Grace, E., & Lesseig, K. (2021). Student ways of thinking in STEM contexts: A focus on claim making and reasoning. School Science and Mathematics, 21 (8), 466–480. https://doi.org/10.1111/ssm.12501 .

Slavit, D., Lesseig, K., & Simpson, A. (2022). An analytic framework for understanding student thinking in STEM contexts. Journal of Pedagogical Research, 6 (2), 132–148. https://doi.org/10.33902/JPR.202213536

Stehle, S. M., & Peters-Burton, E. E. (2019). Developing student 21st century skills in selected exemplary inclusive STEM high schools. International Journal of STEM Education, 6 , 39. https://doi.org/10.1186/s40594-019-0192-1

Strimel, G. J., Bartholomew, S. R., Kim, E., & Zhang, L. (2018). An investigation of engineering design cognition and achievement in primary school. Journal for STEM Education Research . https://doi.org/10.1007/s41979-018-0008-0

Stroh, D. P. (2018). The systems orientation: From curiosity to courage . https://thesystemsthinker.com/the-systems-orientation-from-curiosity-to-courage/

Svensson, M. (2022). Cross-curriculum system concepts and models. In J. Hallström & P. J. Williams (Eds.), Teaching and learning about technological systems (pp. 53–72). Springer. https://doi.org/10.1007/978-981-16-7719-9_4

Teng, B., Wang, S., Shi, Y., Sun, Y., Wang, W., Hu, W., & Shi, C. (2022). Economic recovery forecasts under impacts of COVID-19. Economic Modelling, 110 , 105821. https://doi.org/10.1016/j.econmod.2022.105821

Tornroth, S., & WikbergNilsson, A. (2022). Design thinking for the everyday aestheticisation of urban renewable energy. Design Studies, 79 , 101096. https://doi.org/10.1016/j.destud.2022.101096

Turner, E., Bennett, A., Granillo, M., et al. (2022). Authenticity of elementary teacher designed and implemented mathematical modeling tasks. Mathematical Thinking and Learning: an International Journal, 2022 , 1–24.

Turner, E., Roth McDuffie, A., Aguirre, J., Foote, M. Q., Chapelle, C., Bennett, A., Granillo, M., & Ponnuru, N. (2021). Upcycling plastic bags to make jump ropes: Elementary students leverage experiences and funds of knowledge as they engage in a relevant, community oriented mathematical modeling task. In J. Suh, M. Wickstram, & L. English (Eds.), Exploring mathematical modeling with young learners (pp. 235–266). Springer.

Turner, E., Varley, M., Simic, K., & Diaz-Palomar, J. (2009). “Everything is math in the Whole World!”: Integrating critical and community knowledge in authentic mathematical investigations with elementary latina/o students. Mathematical Thinking and Learning: An International Journal, 11 (3), 136–157.

Tytler R (2016) Drawing to learn in STEM. Proceedings of the ACER Research Conference: Improving STEM learning: What will it take? (pp. 45-50). Australian Council for Educational Research.

van Breukelen, D. H. J., de Vries, M. J., & Schure, F. A. (2017). Concept learning by direct current design challenges in secondary education. International Journal of Technology and Design Education, 27 , 407–430.

York, S., Lavi, R., Dori, Y. J., & Orgill, M. (2019). Applications of systems thinking in STEM education. Journal of Chemical Education, 96 , 2742–2751.

Zawojewski, J., Hjalmarson, M., Bowman, K. J., & Lesh, R. (Eds.). (2008). Models and modeling in engineering education . Brill.

Download references

Acknowledgements

Sentiments expressed in this article have arisen from recent Australian Research Council grants # DP 220100303 and DP 150100120. Views expressed in this article are those of the author and not the Council.

Open Access funding enabled and organized by CAUL and its Member Institutions.

Author information

Authors and affiliations.

Queensland University of Technology, Brisbane, Australia

Lyn D. English

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Lyn D. English .

Ethics declarations

Conflict of interest.

There are no financial or non-financial interests that are directly or indirectly related to this submission.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

About this article

English, L.D. Ways of thinking in STEM-based problem solving. ZDM Mathematics Education 55 , 1219–1230 (2023). https://doi.org/10.1007/s11858-023-01474-7

Download citation

Accepted : 16 February 2023

Published : 03 March 2023

Issue Date : December 2023

DOI : https://doi.org/10.1007/s11858-023-01474-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Problem solving
• Mathematical modelling
• Critical thinking
• Systems thinking
• Design-based thinking
• Adaptive and innovative thinking
• Find a journal
• Publish with us
• Track your research

• [email protected]
• Login / Register

Why Critical Thinking Is Important: Skills and Benefits Explained

Article 07 Sep 2024 49 0

Critical thinking is one of the most essential cognitive skills, vital for navigating today’s complex world. Whether you are a student, educator, professional, or lifelong learner, developing critical thinking abilities can dramatically improve your problem-solving skills, decision-making processes, and intellectual independence.

What is Critical Thinking?

At its core, critical thinking refers to the ability to think clearly and rationally, understanding the logical connections between ideas. It involves analyzing, evaluating, and synthesizing information to form judgments. Rather than accepting information at face value, critical thinkers delve deeper, questioning assumptions and assessing the evidence before drawing conclusions.

Components of Critical Thinking

Critical thinking is a multifaceted skill, which includes:

• Analysis : Breaking down complex information into smaller, more manageable parts.
• Evaluation : Judging the credibility of sources and the strength of arguments.
• Logical reasoning : Making decisions based on coherent and well-structured thought processes.
• Intellectual independence : Forming opinions based on careful examination rather than external influences.

By understanding these components, you can build a strong foundation for developing critical thinking skills , applicable to all aspects of life.

Importance of Critical Thinking in Daily Life

Why is critical thinking important? In today’s information-driven world, where we are constantly bombarded with data, opinions, and perspectives, being able to distinguish facts from fiction is essential. Critical thinking plays a key role in decision-making , problem-solving , and understanding complex issues . Let’s explore its significance in various areas:

1. Decision-Making

Effective decision-making requires the ability to weigh options, consider consequences, and anticipate outcomes. Critical thinking enables individuals to make informed decisions based on facts and reason, rather than emotions or bias. For example, when choosing a career path, a critical thinker will evaluate job market trends, personal interests, and long-term goals to make a well-informed decision.

2. Problem-Solving

In both personal and professional environments, problem-solving is an everyday necessity. Critical thinking enhances your ability to tackle challenges by breaking problems into manageable parts, analyzing potential solutions, and selecting the most effective approach. When faced with a complex issue at work, for example, a critical thinker will systematically evaluate possible solutions rather than jumping to conclusions.

3. Understanding Complex Issues

Whether in academics, work, or social interactions, understanding multifaceted issues requires higher-order thinking . Critical thinking allows you to approach these issues objectively, avoiding oversimplifications or emotional reactions. This skill is particularly valuable in today’s polarized world, where complex social, political, and environmental issues require careful thought and nuanced understanding.

Benefits of Developing Critical Thinking Skills

1. improved problem-solving abilities.

One of the most significant advantages of honing critical thinking is the improvement in problem-solving abilities. By learning to break down issues and consider various angles, you become adept at finding solutions that others might overlook. This skill is particularly valuable in fields like business, technology, and education, where innovation and creativity are prized.

2. Better Decision-Making

Good decisions stem from the ability to evaluate situations clearly and rationally. Critical thinking ensures that you are not swayed by emotional or impulsive reactions but instead base your decisions on sound reasoning. This leads to more consistent, reliable outcomes in both personal and professional settings.

3. Enhanced Communication Skills

Effective communication is built on clear thinking. By developing critical thinking skills , you improve your ability to express ideas clearly and persuasively. Whether in writing or speech, critical thinkers can present their ideas logically and support them with solid evidence, making their arguments more compelling.

4. Increased Intellectual Independence

Critical thinkers develop the habit of questioning information and seeking their own answers. This fosters intellectual independence , enabling individuals to form their own opinions and perspectives based on careful analysis rather than relying on others’ viewpoints. This independence is crucial in a world where misinformation is rampant.

5. Boosted Creativity and Innovation

Critical thinking is not only about logic; it also encourages creativity. By questioning assumptions and exploring new possibilities, critical thinkers often develop innovative solutions and ideas. This is particularly beneficial in fields that require out-of-the-box thinking, such as technology, entrepreneurship, and the arts.

How to Develop Critical Thinking Skills

Developing critical thinking skills is a lifelong process, but with intentional practice, anyone can improve. Here are actionable steps you can take:

1. Question Assumptions

The first step in critical thinking is to challenge your assumptions. Ask yourself: Why do I believe this? What evidence supports this claim? By questioning assumptions, you open your mind to new perspectives and solutions.

2. Analyze Information Critically

When presented with information, whether in the form of news, reports, or advice, practice analyzing it critically. What is the source of the information? Is it reliable? Are there biases that could affect the interpretation of the data? Critical thinking requires that you not only take information at face value but also dig deeper to understand its context.

3. Seek Out Diverse Perspectives

Exposure to different viewpoints strengthens your ability to think critically. By engaging with a wide range of opinions and ideas, you can challenge your preconceptions and develop a more balanced view of complex issues.

4. Reflect on Your Thinking Process

Reflective thinking is an integral part of critical thinking . After making decisions or solving problems, take time to evaluate your thought process. What worked well? Where could you improve? This self-reflection helps refine your thinking skills over time.

5. Practice Problem-Solving in Real-Life Scenarios

A great way to develop critical thinking is by applying it to real-life situations. Take on complex tasks at work, analyze current events, or even engage in strategy games that require planning and decision-making. By practicing problem-solving regularly, you will sharpen your critical thinking abilities.

Teaching Critical Thinking Skills

Critical thinking is not an innate ability; it must be nurtured and developed, particularly in educational settings. Here’s how educators can teach these essential skills:

1. Encourage Open-Ended Questions

In the classroom, asking open-ended questions encourages students to think critically rather than simply regurgitate information. Questions like “Why do you think this is true?” or “How would you solve this problem?” prompt students to analyze information and consider alternative solutions.

2. Promote Active Learning

Engaging students in active learning exercises, such as debates, group discussions, or problem-based learning, helps develop their critical thinking . These activities require students to articulate their thoughts, evaluate others’ perspectives, and revise their ideas based on evidence.

3. Teach Metacognitive Skills

Metacognition, or thinking about one’s thinking, is crucial for critical thinking . Teachers can encourage students to reflect on their thought processes by asking them to explain how they arrived at their conclusions. This practice helps students become more aware of their thinking patterns and biases.

Critical thinking is a powerful skill that enhances decision-making, problem-solving, and intellectual independence. It is essential in today’s fast-paced, information-rich world, where the ability to analyze information and make informed decisions is critical. By developing critical thinking skills , you can navigate complex issues with confidence, communicate more effectively, and innovate in both personal and professional environments.

To enhance your critical thinking , practice questioning assumptions, analyzing information critically, and seeking out diverse perspectives. Whether in education, work, or daily life, these skills will equip you to face challenges with clarity and logic, leading to better decisions and greater personal growth.

• Latest Articles

AI and the Future of Humans: Impact, Ethics, and Collaboration

Why ai is the future: innovations driving global change, how information technology is shaping the future of education, the impact of ai on employment: positive and negative effects on jobs, negative impact of ai on employment: automation and job loss, top 10 benefits of infographics in education for enhanced learning, positive impact of ai on employment: opportunities & benefits, blended learning: maximize impact with hybrid education models, accessibility in education: every student's voice matters, ai’s influence on job market trends | future of employment, improve your writing skills: a beginner's guide, shifts in pop culture & their societal implications, top study tips for exam success: effective strategies for academic achievement, 9 practical tips to start reading more every day, top study strategies for student success, personalized learning: tailored education for every student, how to study smarter, not harder: tips for success, 10 important roles of technology in education, apply online.

Find Detailed information on:

• Top Colleges & Universities
• Popular Courses
• Exam Preparation
• Admissions & Eligibility
• College Rankings

Sign Up or Login

Not a Member Yet! Join Us it's Free.

Already have account Please Login

Pardon Our Interruption

As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen:

• You've disabled JavaScript in your web browser.
• You're a power user moving through this website with super-human speed.
• You've disabled cookies in your web browser.
• A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article .

To regain access, please make sure that cookies and JavaScript are enabled before reloading the page.